Transcript Slide 1

CS110: Introduction to Computer Science – Lab Module 4
Computational Sociology:
Social Choice and Voting Methods
The Process of Computing
Election Victories
Quantitative skills and concepts
Data Analysis
Mathematical Modeling
Algorithms for Rank determination
Rank Aggregation
Prepared by Fred Annexstein
University of Cincinnati
Some rights reserved.
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The Rank Aggregation Problem
“Consensus” ranking of all
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Let us create our own data by ranking the previous 3 labs.
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Submit your rankings on Bb Chat
• submit in your rank order of the three candidates
– Lab 1 - “Napoleon”
– Lab 2 - “Al Gore (Mr. Global Warming)”
– Lab 3 - “Archimedes”
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Voting using Plurality Method
• Plurality method
Election of 1st place votes
• Plurality candidate
The Candidate with the most 1st place votes
• In your worksheet determine the number of 1st
place votes for each candidate
• Is there a Majority candidate?
– A majority candidate has > 50% of 1st place votes
– If not, then is the plurality candidate a good and fair
choice?
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A Fairness Criteria
• Condorcet Criterion: A candidate which wins every
other in pairwise simple majority voting should be ranked
first. A plurality candidate may or may not satisfy this.
• Does the plurality candidate in our election satisfy this
Condorcet Criterion?
• To determine this we need to compute pairwise victors.
1v.2, 1v.3, 2v.3, etc.
• If a candidate wins every head-to-head comparison call it
a Condorcet candidates. Not always possible! Why?
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The Method of Pairwise Comparisons
• The winner of each pairwise comparison gets 1 point
and the loser gets none; in case of a tie each candidate
gets ½ point. The winner of the election is the candidate
with the most points after all the pairwise comparisons
are tabulated.
• Determine the pairwise comparison scores for each of
the three candidates.
• Is there a victorious candidate using this method?
• In our election between 3 candidates, there are 3
pairwise comparison contests.
• How many comparison contests will be needed for an
election having 6 candidates? Can you determine a
formula c(n) for the case of n candidates?
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An Alternative: Borda’s method
• Head-to-head comparisons can get out of control.
• Borda Count Method: an easy “score-based” method.
Each place on a ballot is assigned points. In an election
with N candidates we give 1 point for a last place, 2
points for second from last place, and so on.
• So in our example we give 3 points for 1st, 2 points for 2nd,
and 1 point for 3rd.
• Compute Borda scores for all three candidates.
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An Alternative: Kendall’s Method
• Want to answer question: of all potential orderings, which
is the best?
• Use Kendall tau distance between two ranked lists
– Count the number of pairwise disagreements
between the two lists
• Compute the Kendall Tau distances for all 3!=6 potential
orderings
• This can be done by using data from part 1 on pairwise
contests. For example, for potential candidate ordering
(1,2,3) there are
– 3 disagreements for ordered pair (1,2)
– 3 disagreements for ordered pair (1,3)
– 2 disagreements for ordered pair (2,3)
-> 8 total disagreements for ordering (1,2,3)
• Which of 6 orderings gives lowest (best) score for our
candidate election?
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A Celebrated Theorem
You might be asking yourself whether
there is a method that is superior to all others.
In 1972 Kenneth Arrow won the Nobel Prize in Economics
for his social choice theory.
Arrow’s Impossibility Theorem:
It is mathematically impossible for a democratic voting method
to satisfy a set of natural fairness criteria.
Submit your final worksheet to Blackboard Dropbox.
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